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At this point in a lecture by Cedric Villani, prof. Villani talks about his work revealing that "if entropy is a concave function of time, then the Ricci curvature is non-negative."

But wait, how can entropy be concave in time? Doesn't the 2nd law of thermodynamics insist that entropy is monotonically increasing in time?

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    $\log$ is monotonically increasing, and concave. – 2017-02-24
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    Ah, I see he did not draw a monotonic function. Still, concavity and monotonicity can coexist. – 2017-02-24

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